INTRODUCTION TO POLYNOMIAL CALCULUS
1. Straight Lines
Given two distinct points in the plane, there is exactly one straight line that contains
them both. This is one of the important principles of Plane Geometry. If the plane
is equipped with a Cartesian coordinate system, it should be possible to write down an
equation for such a line in terms of the x and y coordinates. In this section we shall show
how to do this. The notion of slope will be very useful in this task.
The slope of a line is dened to be the change in the y coordinate (the rise) divided
by the change in the x coordinate (the run) as we move from one point on the line to a
second point on the line (see Figure 1.1). That is, if (x1; y1) and (x2 ; y2) are two points
on the line, then the slope of the line is the number m, where
, y1
m = xy2 ,
x1
2
This number does not depend on which two points on the line are chosen. In fact, if two
other points (x3; y3) and (x4; y4) are chosen, then it follows from the similar triangles in
Figure 1.2 that
y2 , y1 = y4 , y3
x2 , x1 x4 , x3
and so the points (x3; y3) and (x4; y4) give the same value for the slope m as do the points
(x1; y1) and (x2; y2).
The slope measures whether a line rises or falls as we move to the right and how steeply
it does so. Positive slope means the line rises to the right, while negative slope means
it falls to the right. Slope zero means the line is horizontal, since it means that the y
coordinate does not change at all from point to point on the line. A vertical line has no
slope (some would say it has innite slope). This is because the x coordinates of points on
such a line are all the same and so the denominator is zero in the equation dening slope.
Example 1. Find the slope of the line which contains the points (1; 2) and (3; 5).
Solution: Here the rise is 5 , 2 = 3 and the run is 3 , 1 = 2 and so the slope is
rise
run
= 3=2.
The point-slope form of the equation of a line. Using slope we can easily write
down an equation for any line that is not vertical. Let m be the slope of the line and let
(x0; y0) be some xed point on the line. If (x; y) is a variable point on the line, then
, y0
m = xy ,
x0
and so
y , y0 = m(x , x0)
1
2
This is called the point-slope form of the equation of a straight line. It gives the equation
of the line in terms of the slope m of the line and a point (x0 ; y0) on the line.
Example 2. Find the equation of the line which passes through the point (1; 3) and
has slope 2.
Solution: Here the slope m is 2 and the point (x ; y ) is (1; 3) and so the equation of
the line is
0
0
y , 3 = 2(x , 1) or y = 2x + 1
Example 3. Find the equation of the line which contains the points (,1; 2) and (0; 5)
Solution: The slope of this line is
m = 0 ,5 ,(,21) = 3
and the line contains the point (0; 5); so the equation of the line is
y , 5 = 3(x , 0) or y = 3x + 5
The slope-intercept form of the equation of a line. If a line crosses the y-axis at
the point (0; b), then the number b is called the y-intercept of the line. If the line has slope
m, then the point-slope form of its equation (using the point (0; b)) is y , b = m(x , 0) or
y = mx + b
This is the point-intercept form of the equation of the line. Note that every line except a
vertical line does cross the y-axis and, thus, has a slope-intercept form for its equation.
Example 4. What is the equation of the line with slope ,5 and y-intercept 4 ?
Solution:
y = ,5x + 4
The general equation of a line. Every equation of the form
Ax + By + C = 0
is the equation of a line and, conversely, every line has an equation that can be put in this
form. A non-vertical line has an equation of the form y = mx + b, which can be written
mx , y + b = 0
3
which is of the form Ax + By + C with A = m, B = ,1 and C = b. On the other hand, a
vertical line has the form
x=k
where k is a constant. This can be re-written as
x,k =0
which is of the form Ax + By + C = 0 with A = 1, B = 0, and C = ,k.
Example 5. Find the slope and y-intercept for the line which is described by the
equation 2x + 3y , 6 = 0.
Solution: If we solve for y in this equation, we get the equation
y = , 32 x + 2
The equation of the line is now in slope-intercept form and we can just read o the slope
(, 23 ) and the y-intercept (2).
Parallel and perpendicular lines. If the two lines in Figure 1.3 are parallel, then the
two triangles are similar, from which it follows that the two lines have the same slope. On
the other hand, if the two lines have the same slope then the two triangles in Figure three
must be similar, since they contain right angles with proportional adjacent sides. This
implies that the two lines are parallel since they make the same angle with a horizontal
line. Thus, two lines are parallel if and only if they have the same slope.
What about perpendicular lines? There is an old carpenter's trick for checking that an
angle is a right angle. One measures o three units along one leg and makes a mark and
then measures four units along the other leg and makes another mark. The angle is then
a right angle if and only if the distance between the two marks is ve. This works because
32 + 42 = 52 and an angle in a triangle is a right angle if and only if
a2 + b2 = c2
where a and b are the lengths of the two agacent sides of the angle and c is the length of
the opposite side (the Pythagorean Theorem).
Let us see how we can apply this to tell when two lines are perpendicular. We may as
well assume that neither line is horizontal, since it is easy to see when a line is perpendicular
to a horizontal line (it must be vertical). Then the rst line intersects the x-axis in a point
(x1; 0) and the second line intersects the x-axis in a point (x2 ; 0). Let's suppose the two
lines are not parallel and so they intersect each other in a point (x0 ; y0) (see Figure 1.4).
Then they intersect at a right angle if and only if a2 + b2 = c2, where a is the distance
from (x0 ; y0) to (x1 ; 0), b is the distance from (x0; y0) to (x2; 0) and c is the distance from
(x1; 0) to (x2 ; 0). This leads to the equation
4
which simplies to
(x1 , x0 )2 + y02 + (x2 , x0 )2 + y02 = (x2 , x1 )2
2x20 + 2y02 , 2x0 x1 , 2x0 x2 = ,2x1 x2
On dividing by 2 and rearranging terms this becomes
y02 = ,x20 + x0x1 + x0x2 , x1x2
or, after factoring,
or
which says that
y02 = ,(x0 , x2 )(x0 , x1)
x0 , x1
=
,
x0 , x2
y0
y0
m2 = , m1
1
where m1 is the slope of the rst line and m2 is the slope of the second line. Thus, two
non horizontal lines are perpendicular if and only if the slope of the second is
the negative reciprocal of the slope of the rst.
Example 6. Find the equation of the line which is parallel to the line with equation
2x + 3y = 5 and passes through the point (1; 2).
Solution: We solve for y in the equation of the rst line in order to put its equation in
the form
y = , 32 x + 53
This shows that the line has slope , 23 . The line which has the same slope and passes
through the point (1; 2) has the equation
y , 2 = , 32 (x , 1) or y = , 23 x + 38 x or 2x + 3y = 8
Example 7. Find the equation of the line which is perpendicular to the line 2y + x = 3
and meets it at the point (1; 1).
Solution: If we solve the rst equation for y it becomes
y = , 12 x + 32
which tells us that its slope is , 12 . A line perpendicular to this line will, therefore, have
slope 2. Since the line we seek must pass through the point (1; 1), its equation is
y , 1 = 2(x , 1) or y = 2x , 1
5
2. Slope of a Curve
We know about the slope of a straight line. It is the change in the y-coordinate divided
by the change in the x-coordinate (rise divided by run) as we move from a given point on
the line to any other point on the line. The law of similar triangles says that this ratio is
independent of the two points on the line that are chosen.
What about the slope of a curve that is not a straight line? Does this make sense? If
so, how do we calculate it? Let's look at an example, say the curve
y = x2
The graph of this curve (Figure 2.1) makes it clear that, if its slope makes sense, it cannot
be a xed number. To the left of x = 0 the curve slopes downward (negative slope) while
to the right of x = 0 the curve slopes upward (positive slope) and it rises more steeply the
further to the right we go. Thus, if it makes sense at all, the slope must depend on where
we are on the curve.
In fact, the slope of the curve y = x2 does make sense at each point (x; y) on the curve
(but it changes as the point changes). This is suggested by the fact that if a microscopically
tiny piece of the curve is magnied enough to be visible then it looks like a straight line.
This is illustrated by the graphs in Figure 2.1 which show the curve near the point (1; 1)
magnied by various factors. In other words, the smaller the segment of the curve we look
at near (1; 1), the more the curve looks like a straight line. The slope of this straight line
should be what we mean by the slope of the curve at the point (1; 1).
How can we calculate this slope? We do the same thing we would do if the curve were
a straight line. We calculate the change in y divided by the change in x as we move from
one point on the curve to another. However, now we choose special points. We let the rst
one be (1; 1) itself and we choose the second one to be near (1; 1). The nearer to (1; 1)
we make it, the more the curve between these two points looks like a straight line and the
closer our ratio will be to the slope of this line. Let's try this for some choices of points on
the curve y = x2 (see Figure 2.2). In each case we will be calculating the rise divided by
the run between a rst point, (1; 1), and some nearby second point on the curve. In other
words we will be calculating the slope of the line joining these two points.
With second point (3; 9) we get
,1 = 8 =4
slope = 39 ,
1 2
With second point (2; 4) we get
1 = 3 =3
slope = 42 ,
,1 1
With second point (1:5; 2:25) we get
, 1 = 1:25 = 2:5
slope = 21::25
5,1
:5
6
With second point (1:1; 1:21) we get
, 1 = :21 = 2:1
slope = 11::21
1 , 1 :1
With second point (1:01; 1:0201) we get
, 1 = :0201 = 2:01
slope = 1:10201
:01 , 1
:01
A graphic description of what is happening here is shown in Fig. 2.
One can now guess that if we continue doing this for second points that are closer and
closer to (1; 1), the slope we are calculating will simply get closer and closer to the number
2. This would mean that the slope of the curve y = x2 at the point (1; 1) is 2.
Let us now take a perfectly general second point { one that is obtained by changing
the x coordinate of the rst point by an amount h so that the second point becomes
(1 + h; (1 + h)2 ). The smaller h is chosen the closer the second point is to (1; 1). The slope
we get for the line joining (1; 1) to this second point is then
2
h2 = 2 + h
slope = (1 + hh) , 1 = 2h +
h
Now if h is chosen very small then this number will be very close to 2. In other words,
as h approaches 0 our slope approaches 2. This should convince us that the slope of the
curve y = x2 at the point (1; 1) is 2.
The fact that 2 + h approaches 2 as h appoaches 0 is commonly written
lim (2 + h) = 2
h!0
Here h!
lim0(2+ h) is shorthand for \the limit of 2+ h as h approaches 0" and it simply means
the number that 2 + h approaches as h approaches 0.
The method used above works for other curves as well.
Example 1. Find the slope of the curve y = x , 3x at the point where x = 2.
Solution: For the function f (x) = x , 3x we have
2
2
f (2) = ,2
f (2 + h) = (2 + h)2 , 3(2 + h) = ,2 + h + h2
Thus, the slope of the curve when x = 2 is
lim0 h +h h = h!
lim0(1 + h) = 1
slope = h!
lim0 f (2 + hh) , f (2) = h!
2
7
Returning to the curve y = x2 , we would like a formula for the slope of this curve at
any point of the curve, not just at the point (1; 1). We use the same technique. Given a
point (x; x2) on the curve, we move to a nearby second point (x + h; (x + h)2 ), obtained by
changing the x coordinate of the rst point by an amount h. Then we calculate the slope
of the line joining these two points. It is
2
2
2
2
xh
+
h
(
x
+
h
)
,
x
= h = 2x + h
slope =
h
As h approaches zero this slope approaches the number 2x. In other words
lim (2x + h) = 2x
h!0
We conclude that the slope of the curve y = x2 at the point (x; x2) is 2x.
The preceding discussion leads to the following denition of the slope of a curve which
is given as the graph of a function f :
Denition A. The graph of a function f is said to have slope m at a point (x; f (x)) on
the graph provided
f (x + h) , f (x) = m
lim
h!0
h
That is, provided the expression f (x+hh),f (x) approaches m as h approaches 0. In this case,
we call the number m the derivative of f at x and denote it by f 0(x).
Thus, the derivative of a function f is another function f 0 of the same variable x. Its
value at x is the slope of the graph of f at the point (x; f (x)). In other words, it is the
instantaneous rate of change of y with respect to x as we pass through the point (x; f (x))
while moving along the graph of f .
For now we will take the statement \ f (x+hh),f (x) approaches m as h approaches 0"
as intuitively understood but later in the course we will need to make this idea more
precise. We will do this when we study limits. As we shall see in the next section, when
f is a polynomial it is quite easy to see what happens to the expression f (x+hh),f (x) as h
approaches 0 and so to study derivatives of polynomials we do not need a sophisticated
study of limits.
The discussion preceding the above denition shows that the derivative of x2 is 2x. We
give two other examples of the calculation of a derivative.
Example 2. Find the derivative of a constant function f (x) = c.
Solution:
f 0(x) = lim f (x + h) , f (x) = lim c , c = lim 0 = 0
h!0 h
h!0
h
Thus, the derivative of a constant function is 0. This just reects the fact that the graph
of a constant function is a horizontal line and, thus, has slope 0.
h!0
8
Example 3. Find the derivative of the function f (x) = 3x , 2.
Solution:
f 0 (x) = lim f (x + h) , f (x) = lim 3(x + h) , 2 , (3x , 2) = lim 3 = 3
h!0
h!0
h
h
Thus, the derivative of the function f (x) = 3x , 2 is the constant 3. This is not surprising,
since the graph of the function f (x) = 3x , 2 is a straight line with slope 3.
h!0
3. Derivative of a Polynomial
In the last section we dened the derivative f 0(x) of a function f (x) to be the slope of
the curve y = f (x) at the point (x; f (x)). This, in turn, is dened to be the number that
the expression f (x+hh),f (x) approaches as h approaches 0. In other words
f 0 (x) = h!
lim0 f (x + hh) , f (x)
We will now use this denition to calculate the derivative of any polynomial. We begin
by calculating the derivative of the monomial xn for each value of n. For n = 0; 1; 2; 3
this was done in the previous section and its problem set. However, we will repeat these
calculations here in order to demonstrate the pattern that emerges.
If f (x) = x0 = 1 then
If f (x) = x1 = x then
If f (x) = x2 then
If f (x) = x3 then
f 0(x) = h!
lim0 1 ,h 1 = h!
lim0 0 = 0
f 0 (x) = h!
lim0 x + hh , x = h!
lim0 1 = 1
2
2
(
x
+
h
)
,
x
0
= h!
lim0(2x + h) = 2x
f (x) = h!
lim0
h
3
3
f 0(x) = h!
lim0 (x + hh) , x = h!
lim0(3x2 + 3xh + h2 ) = 3x2
The pattern here suggests that, for any natural number n, the derivative of xn should be
nxn,1 . This is, in fact, true. The proof is not dicult. From the denition of derivative,
we have that
n n
(xn )0 = h!
lim0 (x + hh) , x
9
If we expand (x + h)n using the binomial theorem we obtain
(x + h)n = xn + nxn,1 h + n(n2, 1) xn,2 h2 + + nxhn,1 + hn
It is not important here to know exactly each term in this expansion. It is important to
know the rst two terms and the fact that every term except the rst two has a factor of
h raised to a power of at least two. When we subtact xn from this expansion and divide
by h we obtain
(x + h)n , xn = nxn,1 + n(n , 1) xn,2 h + + nxhn,2 + hn,1
h
2
Here, every term except the rst one has a factor of h. Thus, when we take the limit as h
appoaches 0, all terms except the rst one will vanish. We conclude that
n n
(xn )0 = h!
lim0 (x + hh) , x = nxn,1
This proves the following theorem:
Theorem A. If n is any non-negative integer, then
(xn )0 = nxn,1
We now know the derivatives of a large number of functions. For example:
(x4 )0 = 4x3
(x10)0 = 10x9
(x95)0 = 95x94
Next we would like to be able to nd the derivative of a polynomial like x3 +4x2 , x +5
which is a linear combination of monomials. We need the following theorem:
Theorem B. If f and g are functions and c is a constant, then
(1)
(2)
(cf )0 = cf 0
(f + g)0 = f 0 + g0
That is, the derivative of a constant times a function is that constant times the derivative of
the function and the derivative of the sum of two functions is the sum of their derivatives.
We won't prove this theorem now. Its proof is in the textbook and will be done later
in the course. Using this theorem and the preceding one, we can now nd the derivative
of any polynomial.
10
Example 1. Find the derivative of 3x + 2x + 5.
Solution:
2
(3x2 + 2x + 5)0 = (3x2)0 + (2x)0 + (5)0
= 3(x2)0 + 2(x)0 + (5)0
= 3 2x + 2 1 + 0
= 6x + 2
Th B(2)
Th B(1)
Th A
Example 2. Find the derivative of x , 11x + 9x + 2.
Solution:
5
3
(x5 , 11x3 + 9x + 2)0 = (x5)0 + (,11x3 )0 + (9x)0 + (2)0
= 5x4 , 11 3x2 + 9 1 + 0
= 5x4 , 33x2 + 9
Example 3. Find the slope of the curve y = x , 3x + 2 at the point (1; 0).
Solution: We rst nd the derivative of x , 3x + 2:
4
4
2
2
(x4 , 3x2 + 2)0 = 4x3 , 3 2x + 0 = 4x3 , 6x
We then evaluate the derivative at x = 1 to get the slope of the curve at the point (1; 0).
Thus, the answer is
4 , 6 = ,2
Example 4. Find the rate of change of the function f (x) = x3 , 4x with respect to x
when x = 2.
Solution: We rst nd the derivative of f (x):
f 0(x) = (x3 , 4x)0 = 3x2 , 4
We then evaluate the derivative at x = 2 to nd the rate of change of f (x) with respect
to x at x = 2. Thus, the answer is
3 22 , 4 = 8
11
Velocity and Acceleration
Much of Physics and Engineering is concerned with the mathematics of moving bodies.
Suppose a body moves along a straight line. If we x an origin for this line and units for
measuring distance on the line, then the position of the body at any time t is described
by its coordinate on the line, often denoted by s(t). The velocity v(t) of the body is then
dened to be the rate of change of s(t) with respect to t - that is, the derivative s0(t) of
s(t). Similarly, the acceleration a(t) of the body is dened to be the rate of change of v(t)
with respect to t - that is, the derivative v0 (t) of v(t). In summary,
v(t) = s0 (t)
a(t) = v0 (t)
Example 5. If a ball if dropped o the top of a 64 foot high building, then its height
s(t) above the ground t seconds later is described by the formula
s(t) = ,16t2 + 64
What is its velocity when it hits the ground? What is its acceleration at any time?
Solution: The ball hits the ground when
,16t2 + 64 = 0:
This happens when t2 = 4; that is, when t = 2. Thus, we need to know the velocity of the
ball when t = 2. But
v(t) = s0 (t) = ,16 2t + 0 = ,32t
At time t = 2 this is ,64. Thus, the ball hits the ground with velocity ,64 ft/sec.
The acceleration at any time t is
a(t) = v0 (t) = (,32t)0 = ,32
Thus, the acceleration is a constant -32 ft/sec/sec.
12
4. Antiderivatives of Polynomials
We have introduced the operation of dierentiation (nding the derivative of a function)
and shown how to dierentiate any polynomial. In this section we will study the reverse
operation.
Denition A. If f (x) is a function, then an antiderivative for f is a function having
f (x) as its derivative.
Antidierentiation undoes or reverses dierentiation. If f is the derivative of g, then g
is an antiderivative of f . Notice that we said \an antiderivative", not \the antiderivative".
This is because a function which has an antiderivative always has innitely many. In fact,
if g is an antiderivative of f then g + c is also an antiderivative of f for any constant c.
This follows from the fact that the derivative of a constant is zero.
Given a function f , does f have an antiderivative? This is a dicult question in general
and will be studied in some detail later in the course. However, for polynomials it is quite
easy to see that the answer is yes. We begin by looking at the monomial xn , where n is a
non-negative integer. Is this the derivative of some other polynomial? If we dierentiate
xn+1 we get
(xn+1 )0 = (n + 1)xn
If we divide by n + 1 this leads to the equation
n+1 0
x
n
n+1 =x
+1
Thus, we have found an antiderivative for xn , namely xnn+1
. This is not the only antiderivative for xn , since we can add any constant to a function without changing its derivative.
Thus, any function of the form
xn+1 + c
n+1
where c is a constant, will have derivative xn .
Denition B. If f (x) is a function, the set of all antiderivatives for f (x) is denoted
Z
f (x) dx
and is called the indenite integral of f (x).
+1
We have shown that every function of the form xnn+1
+ c is an antiderivative for xn .
Conversely, every antiderivative for xn is of this form (as long as we consider the domain
for xn to be the whole real line or, at least, an interval on the real line). However, to prove
this requires one of the big theorems of calculus - the Mean Value Theorem - which will
not be discussed until later in the course. For now we will simply assume that this is true.
Then we can state the following theorem:
13
Theorem A. If n is a non-negative integer, then
Z
n+1
xn dx = nx + 1 + c
where c ranges over all constants.
Remember, this theorem
simply says that the set of antiderivatives for xn is the set of
n
+1
functions of the form xn+1 + c. If we apply this for n = 0; 1; 2 we get:
Z
Z
1 dx = x0 dx = x + c
Z
Z
2
x dx = x1 dx = x2 + c
Z
3
x2 dx = x3 + c
If h is an antiderivative for f and a is a constant, then
(ah)0 = ah0 = af
and so ah is an antiderivative for af . Similarly, if h is an antiderivative for f and k is an
antiderivative for g, then
(h + k)0 = h0 + k0 = f + g
and so h + k is an antiderivative for f + g. Thus, we have proved the following theorem
Theorem B. If f and g are functions and a is a constant, then
Z
Z
(1)
af (x) dx = a f (x) dx
Z
Z
Z
(2)
(f (x) + g(x)) dx = f (x) dx + g(x) dx
In other words, the integral of a constant times a function is the constant times the integral
of the function and the integral of the sum of two functions is the sum of the integrals of
the two functions.
Theorems A and B combined allow us to integrate (nd the indenite integral of) any
polynomial.
Example 1. Integrate x + 3x , 6.
3
14
Solution:
Z
(x + 3x , 6) dx =
Z
Z
Z
x dx + 3x dx + ,6 dx
Z
Z
Z
3
= x dx + 3 x dx , 6 1 dx
2
4
3
x
x
= 4 + 2 , 6x + c
3
3
Th B(2)
Th B(1)
Th A
Note that we can always check our answer to an integration problem by dierentiating
it to see if we get back the original function. For example, let's dierentiate the answer in
the preceding example:
x4 3x2
0 4x3 3 2x
3
4 + 2 , 6x + c = 4 + 2 , 6 + 0 = x + 3x , 6
Indeed, we do get back our original function and this veries that our answer was correct.
Example 2. Find the antiderivative of x , 3x that has value 0 when x = 1.
Solution: All antiderivatives of x , 3x are given by
3
3
Z
2
2
(x3 , 3x2) dx = x4 , x3 + c
4
When x = 1 this becomes
1=4 , 1 + c = ,3=4 + c
Thus, if we want the antiderivative that has value 0 when x = 1, we should choose c = 3=4.
The answer is then
x4 , x3 + 3
4
4
Example 3. The acceleration experienced by an object due to the force of gravity is
,32ft/sec2 . A projectile is red straight up with an initial velocity of 128 ft/sec, after
which the only force acting on it is gravity. What is its velocity t seconds later? When
does it reach its maximum height?
Solution: Acceleration is the rate of change of velocity with respect to time - that is, it
is the derivative of velocity as a function of time. We know the acceleration is ,32ft/sec2.
Thus, the velocity v(t) is an antiderivative of the constant function ,32. So
Z
v(t) = ,32 dt = ,32t + c
15
When t = 0 the velocity is the initial velocity 128 ft/sec. Thus, the constant c must be
128. Therefore, the velocity at time t is
v(t) = ,32t + 128
The object reaches its maximum height when its velocity drops to zero (it is not going up
anymore). This happens when
,32t + 128 = 0
that is, when t = 4 seconds.
Example 4. For the preceding example, assume that the object was red from an
initial height of 10 feet o the ground and then nd the height above the ground of the
object at any time t. Find the maximal height achieved by the object.
Solution: Velocity is the rate of change of distance (in this case height) with respect
to time. That is, it is the derivative of height with respect to time. Thus, the height s(t)
we are seeking is an antiderivative of of the velocity v(t) = ,32t + 128. This means
Z
s(t) = (,32t + 128) dt = ,16t2 + 128t + c
When t = 0 the height is the initial height 10 feet. Thus, c = 10 and
s(t) = ,16t2 + 128t + 10
The maximum height occurs when t = 4 by the previous example. So the maximal height
achieved is ,16 16 + 128 4 + 10 = 266 feet.
5. Denite integrals
There are two matters which we will take as intuitively and operationally acceptable
for the time being. Later in the course we will study these issues at greater depth.
1. If two dierentiable functions f and g have the same derivative in an
interval I , then they dier by a constant. This is the same as saying that if a
function h has derivative equal to 0 everywhere in I , then it is constant (take h = f , g).
Now this is intuitively obvious, if we paraphase it as saying that if a function never changes
anywhere in the interval I , then it is is constant. However, that paraphrase has problems:
the derivative is the instantaneous rate of change, and the assertion: \ a function which
appears not to change at any given instant in fact never changes" , when made precise
loses its intuitive clarity.
16
2. Area is well-dened; regions we shall consider in this course have area.
Again, calculation of areas has been an ongoing problem since the beginning of recorded
history, and the Calculus provides a most eective way of doing this, if we know that it
makes sense. As we shall see below, area of a curvilinear gure is found by approximating
the gure by collections of rectangles. That the approximations lead to a good concept of
area is a subtle problem which for the time being, we shall ignore.
From time to time in this course we will touch on the diculties with these assumptions,
but for the present, we shall assume them as true statements.
Suppose that y = f (x) is a nonnegative function on the interval [a; b]. The area under
the curve y = f (x) from a to b is denoted
Zb
f (x)dx:
a
Theorem. The area under y = f (x) is equal to F (b) , F (a), for any indenite integral
F (x) of f (x) .
Demonstration. For x any point between a and b , let A(x) be the area under the curve
from a to x . Let's calculate A0(x). For a small increment h the area, A(x + h) up to x + h
is approximated by A(x) + hf (x), where hf (x) is the area of the rectangle of base h and
height f (x). Thus
(1)
[A(x + h) , A(x)] = f (x)h + error;
where \error" is bounded by [f (x + h) , f (x)]h, if, for example, f is an increasing
function. Thus
A(x + h) , A(x) , f (x)
h
is bounded by f (x + h) , f (x) which will go to zero if f is at all a reasonable function (we
say, f is continuous). We conclude
A(x + h) , A(x) = f (x):
lim
x,>0
h
Thus A(x) is an indenite integral of f . If F is any other indenite integral,
(2)
A(x) , F (x) = C;
where C is some constant which we can easily nd since A(a) = 0. Substituting in (2) we
obtain C = ,F (a). Thus, the area under the curve from a to b is A(b) = F (b) , F (a) .
Examples
17
1. Find the area under y = xn from 0 to 1.
1
1
xn+1 , so the area is F (1) , F (0) = n+1
.
An indenite integral is F (x) = n+1
2. Find the area under the curve y = x2 + 2x3 from 2 to 4.
Z4
(x2 + 2x3 )dx = 31 x3 + 2 14 x4 j42 =
2
= 31 64 + 12 256 , 13 8 , 21 16 = 416=3:
The denite integral will be dened later in this course as the limit of a sum, as follows.
Suppose that y = f (x) is dened over the interval [a; b]. Let us subdivide the interval by
a sequence of partition points fxi g:
x0 = a < x1 < x2 < < xn,1 < xn = b
Consider the sum
f (x1)(x1 , x0 ) + f (x1)(x2 , x1 ) + + f (xn)(xn , xn,1 )
which can be written, using the summation notation
n1 f (xi)(xi , xi,1 ):
If, for example, we were calculating the area under the graph of a positive function, this
would be the area of a polygonal gure approximating the curvilinear gure, and would
thus give an approximate value for the area. If we divide the gure more nely, using more
points fxi g, we would expect to get a better approximation to the area. Under suitable
conditions (the continuity of the function f ), these sums do approach a limit which is the
denite integral of the function f from a to b.
The Fundamental Theorem of the Calculus will tell us that the denite integral
can be computed as F (b) , F (a), where F is any indenite integral of f .
Example. Suppose a particle moves in a straight horizontal line so that its velocity
directed to the right at time t is v(t) = t , t meters per minute. How far to the left or
2
3
right of the initial position of the particle will it be after 2 minutes?
Answer:
Z2
Z2
4:
v(t))dt = (t2 , t3)dt = 31 t3 , 14 t4j20 = 83 , 16
,
0
=
,
4
3
0
0
the particle is 4=3 of a meter to the left of its initial position.